Whisk the crushed tomatoes, olive oil, minced garlic, and salt together in a bowl until combined; set aside.

Whisk the heavy cream, parmesan, and pepper in a separate bowl until well mixed.

Adjust the oven rack to the middle position and heat the oven to 375°F. Line a large plate with a double layer of coffee filters and spray with vegetable oil spray. Toss the eggplant with salt and spread it evenly over the filters, then microwave on high until the eggplant is dry to the touch and slightly shriveled, about 10 minutes. Let the eggplant cool slightly, then toss it with the zucchini and yellow squash.

Combine oil, minced garlic, and thyme in a small bowl. Heat a portion of the oil in a large nonstick skillet over medium-high heat until shimmering. Add half of the eggplant-squash mixture with salt and pepper and cook, stirring occasionally, until the vegetables are lightly browned, about 7 minutes. Clear the center of the skillet, add half of the garlic-thyme mixture, and cook, mashing with a spatula, until fragrant before stirring everything together and transferring to a bowl; repeat with the remaining oil and vegetables.

Heat the remaining oil in the skillet and cook the spinach until wilted, about 3 minutes. Drain the spinach on a paper towel-lined plate for 2 minutes to remove excess moisture, then stir it into the cooked vegetables.

Grease a 13 by 9-inch baking dish. Spread a portion of tomato sauce over the bottom and arrange a layer of four noodles on top. Spread half of the vegetable mixture, followed by half of the olives, half of the cream sauce, and a portion of mozzarella. Repeat the layering once with noodles, tomato sauce, vegetables, olives, cream sauce, and mozzarella. For the final layer, arrange the remaining noodles on top, cover completely with the remaining tomato sauce, and sprinkle with the remaining mozzarella.

Cover the dish tightly with aluminum foil sprayed with oil and bake until the edges are just bubbling, about 35 minutes, rotating the dish halfway through. Remove from the oven and let the lasagna rest for 25 minutes to allow the layers to set properly. Sprinkle with fresh basil and serve warm.